UNIVERSITA DEGLI` STUDI DI NAPOLI“FEDERICO II” DIPARTIMENTO DIINGEGNERIAELETTRICA E DELLETECNOLOGIE DELL’INFORMAZIONE
DOTTORATO DIRICERCA IN
INGEGNERIAELETTRONICA E DELLETELECOMUNICAZIONI
N
ON
-
LOCAL METHODS
FOR
I
N
SAR
PARAMETERS ESTIMATION
F
RANCESCOPAOLOS
ICAIl Coordinatore del Corso di Dottorato Il Tutore
Ch.mo Prof. Daniele RICCIO Ch.mo Prof. Giovanni POGGI
Il co-tutore
Dr. Ing. Gianfranco FORNARO
Introduction 1 1 Synthetic Aperture Radar 9
1.1 Overview . . . 9
1.2 Geometric resolution . . . 13
1.2.1 Range resolution . . . 14
1.2.2 Azimuth resolution . . . 18
1.3 Single Look Complex statistic . . . 24
2 SAR Interferometry 31 2.1 Overview . . . 31
2.2 Across-track Interferometry . . . 33
2.2.1 Double differences . . . 38
2.2.2 System sensitivity and critical baseline . . . 40
2.2.3 Height of ambiguity . . . 42
2.2.4 Spectral shift . . . 42
2.3 Differential and along-track Interferometry . . . 44
2.3.1 Multi-pass Differential Interferometry . . . 48
2.4 Interferometric phase statistic . . . 52
2.4.1 Phase difference noise model . . . 52
2.4.2 Interferometric phasor noise model . . . 56
3 InSAR parameter estimation 59
3.1 Overview . . . 59
3.2 Maximum-Likelihood Estimation . . . 63
3.3 Dealing with non-stationarities . . . 67
3.3.1 Filter in the spatial domain . . . 67
3.3.2 Filters in the transformed domain . . . 70
3.4 The Non-Local approach . . . 72
3.4.1 NL-InSAR and NLSAR . . . 75
4 Non-Local Estimation in Multipass InSAR techniques 79 4.1 Overview . . . 79
4.2 AML in Multipass InSAR . . . 82
4.2.1 Prior despeckling . . . 86
4.2.2 Interferogram Filtering . . . 87
4.2.3 Distance Measures . . . 88
4.3 Experiments on simulated data . . . 91
4.4 Experiments on real data . . . 98
4.4.1 Results on low-resolution data . . . 102
4.4.2 Results on high-resolution data . . . 114
5 Non-Local LLMMSE estimation for single-pair InSAR 121 5.1 Overview . . . 121 5.2 BM3D . . . 122 5.3 InSAR-BM3D . . . 127 5.3.1 Signal model . . . 127 5.3.2 Noise decorrelation . . . 128 5.3.3 Grouping . . . 129 5.3.4 Collaborative filtering . . . 131
5.3.5 Noise variance estimation . . . 133
5.4 Topography-based similarity measure and filtering . . 135
5.6 Experimental results on real data . . . 154
6 Conclusion 161
Latest Remote Sensing technologies are having great impact on both research and applications. Differently from some years ago, when technological development was achievable only for large institutions, like space agencies, today the number of private companies develop-ing and operatdevelop-ing their own acquisition systems is increasdevelop-ing quickly. This trend is also due to the fact that the imaging system technology is becoming mature allowing for incremental improvements regarding performance and low-size/weigth implementation. A further enabling factor has been the evolution of image processing methodologies, with ever more powerful and effective tools, which made possible the devel-opment of new systems and even the creation of features and applica-tions that were not imagined at the time of the system realization. There are many remote sensing modalities, the most popular of which are related to optical, hyperspectral and radar sensors, each of them with specific peculiarities and purposes. For example, optical sensors have by design a higher resolution with respect to radar sensors due to their operative range frequency. But for the same reason they suffer from possible atmospheric occlusions, while the radar sensors do not. Among the radar systems, the Synthetic Aperture Radar (SAR) has experienced an increasing popularity in the last decades, especially for Earth Observation purposes. Since the first missions, it demon-strated its effectiveness in providing reliable information about the
Earth’s surface such as topography, morphology, roughness and di-electric characteristics of the backscattering layer. In fact, SAR sys-tems demonstrated to possess many useful features, that made the difference compared to competitor systems. Since they operate in the microwave frequency range and provide their own illumination, SAR systems can operate almost independently of meteorological condi-tions and sun illumination. Geometric resolucondi-tions in the order of some meters are achieved with physical antennas of limited size, thanks to the synthetic aperture concept and to pulse compression techniques. The price to pay for these desirable properties is in the high trans-mit power, the considerable amount of signal processing required and, compared to optical imagery, the “unconventional” imaging geometry, as it will be shown in chapter 1.
Spaceborne remote sensing systems travel around the Earth over polar orbits at an altitude going typically from 200Km (space shuttle) to 800Km(satellites) at inclinations ranging from 57◦to 108◦. The SAR system ENVISAT, for example, has a spatial resolution in the order of 5m in azimuth and 25m in ground range, while most recent systems as TerraSAR-X and COSMO-SkyMed can reach a resolution of few meters. The imaged swath is about 50−100Km wide in standard imaging mode and up to 500Km with ScanSAR systems.
SAR Interferometry (InSAR) is today one of the most used techniques exploiting SAR imagery. Any image is a bidimensional representation of some features of interest. SAR interferometric images, in particular, measure the height information over the scene. They are generated by processing a pair of SAR images acquired from slightly different points of view, giving rise to the so called Digital Elevation Map. As it will be widely shown in chapter 2, the height information is codified in the interferometric phase, i.e. the phase difference between the
a tool for detection and mapping of terrain displacements over wide areas, with a precision comparable with fractions of the operative wavelength: millimeter range in case of X-band SAR. This is of great importance for risk management as in case of earthquakes or volcanic activity, for glaciology and ice sheet monitoring, for studying tectonic processes and monitoring ground subsidence due to mining, gas, water and oil withdrawal, etc..
The use of spaceborne SARs as interferometers became popular only in the last decades, although the basic principle dates back to the early 1970s. However, it was only in the early 90s that the first results on terrestrial applications were published. Before 1992 SAR data were only available to a limited group of scientists, the required data elaboration was expensive, and appropriate facilities existed only at few research centers. Interferometry data-sets became available to a larger set of researcher thanks to the ESA satellite ERS-1 launched in 1991. It provided a huge amount of SAR data-sets and a series of research groups began to investigate the method intensively and with success [3].
On July 08, 1993, the journal Nature showed to a wide public the
first differential interferogram representing the deformation induced by the earthquake that affected the Californian area of Landers on June 28, 1992. The interferogram was produced by combining the pair of ERS-1 SAR images taken before (April 24, 1992) and after (August 7, 1992) the seismic event. This event was claimed as the first picture of an earthquake.
The ERS-1 satellite was followed by the twin ERS-2 in 1995. They weren’t the first radars to introduce data suitable for interferome-try, but they achieved the real breakthrough in SAR interferometry. The reason of their success was primarily due to the excellent results
Figure 0.2: Cover of the journal Nature showing the first-ever image
of an earthquake.
achieved in the area of repeat-pass interferometry. In fact the satellite orbit was very stable and it was determined with cm accuracy, the
baseline control was very good and many orbit pairs met the base-line conditions for repeat-pass interferometry. ERS-2 was identical to 1, had the same orbit parameter, and continued the ERS-program with the 35-days repeat period. This mission provided out-standing InSAR performance in the TANDEM mission where ERS-1 and ERS-2 were operated in parallel, reaching high coherence orbit pair. In fact ERS-2 followed ERS-1 on the same orbit at a 35 min delay thus, considering Earth’s rotation, this orbit scenario assured that ERS-1 and ERS-2 imaged the same areas at the same look angle at a one day time lag. The orbits were deliberately tuned slightly out of phase such that a baseline of some hundreds meters allowed for cross-track interferometry. This separation between 1 and ERS-2 could be kept very stable, because both satellites were affected by similar disturbing forces. The one day time lag was the key feature to
obtain high coherence interferograms and no other repeat-pass system could achieve it.
Nowadays, the latest SAR systems work in the X-band of the elec-tromagnetic spectrum, that is with a wavelength of about 31mmand provide images with a resolution of up to 1mwith revisit time of a few days. Today the major operational X-band SAR systems are the ger-man TerraSAR-X and the italian COSMO-SkyMed. The former was launched on June 15, 2007 and has been in operational service since January 2008. As for ERS systems a second sister satellite TanDEM-X was launched in early 2010 and nowadays the two satellites act as a pair with the aim of Earth’s Digital Elevation Model (DEM) recon-struction. The latter, COSMO-SkyMed, is a constellation composed of four satellites equipped with SAR sensors. The first satellite of COSMO-SkyMed constellation was launched on June 2007 whereas the full constellation is operational since 2010.
With SAR technology development, the need for digital signal pro-cessing and enhancement tools grew fast. One of the first application was the despeckling: the procedure of removing the speckle from the
amplitude of the SAR images. Due to the presence of many single scat-tering mechanisms in the SAR resolution cell, the backscattered field results in a coherent superimposition of complex phasors that gives rise to constructive and destructive interference phenomena. From one resolution cell to the adjacent, the interference mechanism can vary completely implying a spatial amplitude variation even for areas with constant backscattering characteristics. The speckle phenomena is normally indicated as pseudo-noise, since it is a disturb that impairs the measured signal quality, but, at the same time, it is intrinsic of any coherent measurement system. In SAR images, the speckle impairs the visual interpretation and the performance of subsequent processes like image segmentation and classification. Many despeckling
algo-rithms have been proposed in the last decades [19, 38, 39, 53, 59, 62– 64, 70, 71, 84].
In SAR Interferometry, the enhancement of the interferometric signal is a preliminary step for almost any application. The disturb in this case is mainly given by the degree of dissimilarity between the two im-ages. The amplitude of the normalized complex correlation between
the pair, thecoherence, measures the degree of similarity between the
two images while its argument is the interferometric phase. The es-timation of the correlation allows to obtain first an eses-timation of the phase itself and secondly a measure of the reliability of the estimated phase through the coherence parameter. The classical approach for estimating the complex correlation index ia a moving-average filtering, which, assuming that the interferometric phase has slow spatial vari-ations or, from a statistical point of view, it is locally a Wide-Sense-Stationary (WSS) process, turns out to be the Maximum-Likelihood estimator [95]. Such a filter has to trade off the low estimation vari-ance, achievable with a large window, with image resolution. The res-olution loss is due to a mix of heterogeneous contributions which takes place whenever the signal is not locally WSS as in the case of bound-aries between regions with different backscattering characteristics, e.g. the edge between a field and a street. The work in InSAR parameter estimation has then mainly focused on improving this trade-off by try-ing to achieve a strong filtertry-ing intensity also in non-stationary areas. Back in the ’90s, the Lee [67] and Goldstein [49] filters provided two different solutions working in the spatial and frequency domain, re-spectively. The former adapts the shape of the filtering window to the local phase edges. The latter, instead, enhances the useful spectrum components, assuming that the non-stationary signal is defined in a narrow band while the noise spreads over the whole spectra. The so-lutions proposed during last decades can be divided as well in spatial
and transform domain approaches. The state-of-the-art filters [20, 22] extend the ML estimation to a weighted average in the framework of the nonlocal filtering. This paradigm, rather than using the geo-metrically closest pixels as in the moving average, selects the pixels that are more similar to the target pixel enforcing in this way the sta-tionarity hypothesis on the average group. In [20, 22] new similarity measures are defined to deal with the InSAR statistic and perform then a Weighted-ML estimation.
In this thesis work the nonlocal paradigm has been investigated in the framework of Multitemporal SAR Interferometry, e.g. Differen-tial Interferometry, Tomography, etc., and single InSAR pair, e.g. DEM generation. In the former, Adaptive Multi-Looking methods have been developed for the generation of interferometric data-stacks. Following the nonlocal approach, the proposed methods rely only on similar pixels according to a suitable similarity measure that exploits the stack’s temporal information. An hybrid approach that jointly uses the nonlocal paradigm and transform domain filtering has been investigated for InSAR pair phase estimation. On the track of the BM3D [16] and SARBM3D [84] algorithms, different approaches to the filtering in the transform domain are investigated. Furthermore, a novel approach to the similarity computation and filtering, based on a relative-topography content of the interferometric phase rather than its absolute value, is proposed.
The thesis is organized as follows: chapter 1 introduces to the SAR acquisition system, its resolution and main working principles. In chapter 2 the basic concepts of Interferometry are provided, together with its geometry and its various applications. The problem of In-SAR parameter estimation is described in chapter 3, presenting several state-of-the-art filters. Chapter 4 and Chapter 5 describe the research work carried out on InSAR parameter estimation in the context of
Multitemporal and single-pair Interferometry, respectively. Finally, in chapter 6, the main results are summarized and commented, and future work is outlined.
1.1 Overview
The Synthetic Aperture Radar (SAR) is a system that operates in the microwave domain at the aim of acquiring images of the terrestrial sur-face. The SAR is an active system, in the sense that it provides its own source of radiation, meaning that it can acquire images during both day- and night-time. The microwave operative range lend to the system the ability of watching through clouds and other atmosferic disturbs, since these are practically invisible at this frequency range. The ability of acquiring independently from sunlight and weather conditions made the SAR being very attractive for remote sensing pourposes. During last decades the SAR became a wide spreaded tool complementary to others as the optical and multi/hyper-spectral imaging systems. Clearly, there are other important characteristics as the coherent receiving of the backscattered signal: the capability
of mesuring the signal in its amplitude and phase. The phase in-formation allows, from at least a pair of acquisitions, to measure the altitude of the imaged area (Digital Elevation Model - DEM) or detect
ground deformation over time of the order of centimeter (Deformation Monitoring)[14]. This applications, that are named respectively In-terferometry and Differential Interferometry,will be shown in details
In the following, with reference to Fig. 1.1, the simplest SAR acqui-sition geometry is depicted in order to show the basic SAR working principles and its prime characteristics. The radar is carried on a moving platform that can be an aircraft or a satellite moving at speed V in a straight trajectory at constant altitudeH. The commonly used SAR acquisition mode is the strip-map mode as the one in Fig. 1.1, but other SAR mapping modes are of interest: Spotlight and ScanSAR, that differ from the standard strip-map respectively for the improved resolution and for the wider achieved coverage. In addition there is the most recent TopSAR mode that tries to trade off the benefits coming from the last two mentioned modes. In this context only the standard strip-map mode is used to present the SAR working principles.
Figure 1.1: SAR imaging geometry [3].
The sensor is moving along its assumed straight path, i.e. theazimuth
direction and it sends pulses at its pulse repetition frequency (prf)
and receives coherently the echoes scattered from the Earth’s surface. Each transmitted pulse sweeps across the swath at the velocity of light c and simultaneously the scene is scanned along the azimuth direction at the radar flight speed V. Since the two timescales differ
of several grade of magnitude, the two mechanisms can be considered indipendently. The start-stop approximation is usually assumed: the
radar occupies only some positions in space along its flight path, when the radar sends the pulse it is considered motionless: it waits to receive the scattered echo and then move to the next position. This simplifies the analytical system description and suggests arranging the received echoesside-by-side to form a raw data matrix. The raw data acquired
by a coherent radar resemble to an hologram rather than an image and hence, require a considerable amount of signal processing for image formation, these procedures are indicated in one term asfocusing. The
coordinates of the 2D focused SAR image areranger for the distance of the scattered object from the radar andazimuth xfor the position of the scatterer along the sensor path. The value of the SAR image in the pixel (x, r) depend from the points belonging to the observed scene at coordinates (x0, r0) through the following expression:
u(x, r) = ¨ 4x4R u x0, r0 eiφ(x0,r0)sinc x0−x sinc r0−r dx0dr0 (1.1) where4x and4R are respectively the azimuth and range resolution cell dimensions.
The geometric resolution depends on the acquisition geometry, the
working parameters and the acquisition mode and it is a prime char-acteristic of an imaging system, so that different systems are in first approximation often compared only by that parameter. In the follow-ing the general expressions of the range and azimuth resolutions will be derived for a satellite based SAR, working in strip-map mode. In order to better understand the measured backscattered field in 1.1,
an ideal system with infinite bandwidth is considered: the two sinc
functions become dirac pulses. The 1.1 become:
u(x, r) =˜4x4Ru(x0, r0)eiφ(x0,r0)δ(x0−x)δ(r0−r)dx0dr0 =
=|u(x, r)| ·ejφ(x,r)=M(x, r)·ejφ0·ej4λπr
(1.2)
The amplitude, renamed asM =|u(·)|,of the received signal is related to the amount of energy that is backscattered from the scene and depends on the dielectric behaviour of materials as well as on their physical properties. The phase instead is the sum of three distinct contributions:
• the two-ways travel path: sensor-target-sensor;
• the electromagnetic interaction between the incident electro-magnetic (e.m.) waves and the scatterers within the ground resolution cell;
• the phase shift induced by the processing system used to focus the image.
In particular, referring to eq. 1.2, the term φr = 4λπr is the two-ways
travel path, while the termφ0 accounts for the last two contributions. Generally of a single SAR acquisition (Single Look Complex - SLC)
only the amplitude is of interest, since the phase, as it will be depicted in the following, don’t bring information, but if interferometric appli-cations are the aim, the phase plays an important role. The term φ0, that depends only from the acquired scene, is normally supposed to be the same for the two interferometric acquisitions and hence can be neglected. The difference of theφrterms is related to the two different
travel path associated with the two acquisitions and hence it is used to retrieve information about the topography and/or the displacement affecting the imaged area.
1.2 Geometric resolution
As seen till now, the SAR system is able to identify a point in a bi-dimensional range-azimuth coordinate system. The range direction
is the one that connects the sensor to the target and it is identified by the radar look-angle. This direction is indicated as slant-range
to differenciate it from the ground-range direction that corresponds
to its projection on the ground surface. This second definition is introduced since it provides an handier and more intuitive dimension to describe the imaged targets ground displacement. The azimuth
direction instead corresponds to the fligth path and is the direction along which the SAR antenna is synthesized.
This kind of 2D vision, also if doesn’t allow to separate targets that are at the same distance from the sensor, suits well the problem of a terrestrial surface sensing since the scattering is supposed to come from only one direction: the hypothesis that only a superficial scat-tering mechanism is present. The image distortions due to the radar acquisition aand its geometry are a crucial issue in SAR imaging but are not the aim of this paragraph.
The geometric resolution is defined as the minimum separation (in
range or in azimuth) between two points that can be distinguished as separate by the system [14]. The resolution cell is then given by the
slant-range and azimuth resolutions and it is the smallest dimension that the system can sense. If one considers the ground-range measure in place of the slant-range one, in first approximation and without
con-sidering oversampling processes needed in the image formation step, it can be possible to identify the image’s pixel with the resolution cell.
1.2.1 Range resolution
Refering to Fig. 1.1 it is assumed that the radar beam is directed perpendicular to the flight path of the SAR antenna and downwards at the surface of a flat earth, pointing with a look angleθ respect to thenadir direction. The antenna’s dimensions are indicated withWa
and La respectively for the (y,z) cross section (width) and the (x,z)
cross section (length). In figure Fig. 1.2 is represented the cross section
of the acquisition system geometry depicted in Fig. 1.1 in the plane
(y,z). This view will allow us to retrieve the slant-range resolution
from the acquisition geometry.
Figure 1.2: Cross section: (y, z) plane
The antenna’s beamwidth in the (y,z) cross section is indicated as vertical beamwidth θV, it depends from the antenna’s width Wa and
the working wavelength λ through the relationship: θV = Wλa. From
this parameter depends the coverage area of the radar, also known as ground swath and here indicated with Wg. Since in all the real
systems usually Waλthe hypothesis of small angles is reasonable:
θV 1. Let Rm be the slant-range distance between the radar and
the midswath point on the ground, further noticing that η, the angle formed between the midswath ray and the normal to the terrain sur-face, is the same of the look angleθ, from the system geometry comes that:
Wg ∼=
λ·Rm
Wacosθ (1.3)
In order to have an idea about the dimension of the involved quanti-ties, the ENVISAT SAR system is taken as a reference. The ENVISAT antenna has the following characteristics:
Wa= 1m Rm'800km λ= 5.6cm θ= 25°
Substituting these parameters to the previous formula, it resultsWg'
50km.
For acontinous wave(CW) radar, i.e. a continous waveform is
trasmit-ted, the two targets would be separated only if they are not illuminated by the same beam, hence Wg is identified with the ground-range
res-olution leading then to an unreasonable sres-olution. This is the reason of why the SAR uses a transmission of short monochromatic pulses. In this way the resolution is related to the time duration of the pulse and not uniquely to the acquisition geometry. Indeed, let τ be the time extent of the radar pulse, two targets are distinguishable if their echoes are not superimposed, this is quite a less restrictive situation with respect to the continous wave transmission. The minimum sepa-rations of two resolvable points, respectively in slant and ground range
are then expressed analitically as: 4Rs= c·τ 2 = c 2B (1.4) 4Rg = 4Rs sinθ = c 2B·sinθ (1.5)
wherecis the speed of light andB is the pulse bandwidth: B ≈1/τ . The slant-range resolution is then directly proportional to the pulse duration. The smaller is theτ value the better is the resolution, but this behavior should be traded with an energy constraint: the required pulse durationτ may be too short to deliver adequate energy-per-pulse to produce a sufficient echo’sSignal to Noise Ratio (SNR) for reliable
detection. In particular the smaller is the pulse duration and the higher would be the peak value of the trasmitted power, this leads to solutions that are not feasible in practice.
A pulse compression technique is normally employed to achieve both
high resolution and high SNR. As in traditional radar systems the signal trasmitted by a SAR is a linear chirp modulated pulse.
Figure 1.3: Chirp signal
fre-quency varies linearly with time: s(t) =Acos (2π(f0+αt)t+ϕ0)rect t τ (1.6) The amplitude of s(t) is depicted in Fig. 1.3.
Particularly, the chirp pulse has the following istantaneous frequency:
fist=f0+ α 2π ·t t −τ 2; τ 2 (1.7) where f0 is the frequency at time t = 0 and α is the increasing (or decreasing) chirp rate.
Depending from its basic parameter α it is an up-chirp (α >1) in which the frequency increase with time or otherwise a downchirp (α <1).
Thus the chirp bandwidth can be approximeted as:
Bc∼=4f =ff in−fin= α·τ 2π (1.8)
and the resolution is:
δRs=
c
2Bc (1.9)
The bandwidth of a chirp pulse and its duration are proportional: ατ = 2πB. From the relation in 1.4, in order to achieve better resolu-tion, the pulse bandwidth can be increased by increasing its duration.
In this way the trade-off between pulse bandwith and transmitted energy is overtaken. The radar system range resolution is then deter-mined by the type of pulse coding and the way in which the return from each pulse is processed. The slant-range resolution for the EN-VISAT system is of δRs = 20m and it is achieved with a pulse of
bandwidth B = 160M Hz andτ = 2.7·10−5.
1.2.2 Azimuth resolution
The capability of distinguish two adjacent targets in the fligth path direction is calledazimuth resolution. At the aim of deriving its
ana-litical expression in dependance from the working parameters and the acquisition geometry, the cross section in the (x,z) plane is consid-ered. In practice, given the side-looking configuration, the midswath, the direction connecting in a straigth trajectory the radar sensor with the target, is taken as reference direction in place of the z-axis. This lead to a coordinate system often indicated asazimuth-rangeas shown
in Fig. 1.4.
Figure 1.4: Cross section: azimuth-range
considered in the azimuth resolution formula derivation is the antenna dimension in the considered cross section. The radar antenna in the azimuth dimension was previously indicated with La, then the radar
beam has an angular spread in that dimension of θH = Lλa, where λ
is the wavelenght of the trasmitted signal. The orizontal beamwidth θH determines the azimuth footprint (X), in fact, considering that
usually La λ and hence θH 1 as for the range case, it results
that for a given range distanceRto the target, the azimuth footprint
is:
X=R·θH =
R·λ
La (1.10)
Two objects at the same range distance R can be distinguished in azimuth only if they are in different radar beams, hence the azimuth resolution can be identified with the azimuth footprint ∆x = X. A
system whose resolution is influenced by the actual antenna dimension is called Real Aperture Radar (RAR).
It is worth noting that the azimuth footprint depends from R, hence
different resolutions values are obtained for near and far range: R1 and R2 in Fig. 1.2 respectively. This resolution, with the ENVISAT antenna lengthLa= 10m and for R=Rm, results: ∆x = 5km.
To improve the along-track resolution at some specified range distance R and wavelenght λ, it is necessary to increase the antenna length in the along-track dimension, but this has strong limits. In fact, apart the problem of placing a big antenna on a flying device, the are me-chanical limitation in its construction. Obtaining a surface precision accurate to within a fraction of the wavelength is as more difficult the bigger is the antenna dimension so that a value of La
λ greater than a
few hundred is hard to achieve [14].
order of ∆x = 10m had required the use of an antenna of length
La= ∆Rλx '4.5Km.
Figure 1.5: Synthetic Aperture principle
The basic principle that leads to the use of SAR sensors relates the possibility of simulating an antenna array with just one small sized antenna moving on a straight trajectory. Refering to Fig. 1.5, let the sensor assumeN different positions with a constant stepdwhile mov-ing from A to B, as in the start-stop approximation. Then the point
P is seen N times from the antenna. This behavior is the same of a N-antennas array, in other words the overall system behaves as a longer antenna of length:
L=N ·La (1.11)
The resultant antenna goes under the name of Synthetic Aperture.
The N acquisitions of point P are then combined in a process called
beamforming at the aim of separating the contributions coming from
the point P to the one coming from a pointP’ located at a direction
shown in ref_fig.
Figure 1.6: Synthetic Aperture principle
The hypothesis of plane and monochromatic waves suffices for deriv-ing the formula of the azimuth resolution. The 2-way gain of the synthetic antenna is obained summing all the contributions from the N transmissions: |G(β)| |G(0)| = 1 N sinπ2λLsinβ sinπλ2sinβ·d (1.12) that under the realistic hypothesis of small angles β, the 1.12 can be approximated as: |G(β)| |G(0)| = sinc 2L λ sinβ (1.13)
Since the antenna is discretized in a finite number of radiative ele-ments, the radiating pattern is periodic of frequency:
fx = 2
λsinβ = n
d nZ (1.14)
In order to avoid the replicas superimposition, the correct sampling frequency, and henced, should be find. If a sampling frequency greater
than the Nyquist limit is considered we obtain:
d < 1 2|fx|=
λ
4 (1.15)
that implies the following lower limit on the pulse repetition frequency:
prf > 4
λ·V (1.16)
This last consition cannot be achieved since it wouldn’t guarantee a condition of non-ambiguity in the range direction: if trasmitted pulses overlap than range ambiguity arises.
Hence the last condition should be relaxed using directional antennas (so that ∆β < Lλ
a) that limits the illumination cone only to those
con-tributions coming from the closest angular directions. This works as an anti-alias filter needed because a frequency lower than the Nyquist limit is used. Hence it comes out a newprf lower bound that, jointly
with the rangeprf upper bond, become:
2V La
< prf < c
2δR (1.17)
The operative prf should be chosen in the above range.
The directional antenna of length La limits the number of times the
point P is seen from the radar, and hence the length of the Synthetic Aperture, to:
L=Rm∆ψ=Rm
λ
La (1.18)
Under this condition, the highest prf that respects the conditions in
1.17 leads to the best azimuth resolution achievable while having range non-ambiguity: ∆x≥Rm λ 2L =Rm λ 2 λ La = La 2 (1.19)
The antenna’s actual dimension influences also in this case the az-imuth resolution δx = L2a, but now in the opposite way. The smaller
is the antenna size the better is the resolution. It is worth noting that the azimuth resolution for a synthetic aperture radar is independent from R and a constant value for near and far range points.
Sub-stituting the ENVISAT antenna length in the last formula it results δxEN V ISAT = 4m.
1.3 Single Look Complex statistic
In view of SAR image processing a statistical description for the SAR image, the Single Look Complex (SLC), is provided in the following. In the previous paragraph, the image formation process has been de-scribed as well as the system resolution. Having identified a pixel with the resolution cell, it is clear that a pixel value should account for all the scattering mechanisms that happen within the resolution cell. The possible scattering mechanisms are related to the kind of scatter-ing: superficial scattering, when only an interaction with the surface is supposed, or volumetric scattering, when the wave penetrate the ground and the scattering is seen as a radiating mechanism coming from the interested volume. Scattering mechanism interests the phisi-cal and electromagnetiphisi-cal properties of the target and may be quite vary and complex.
The electromagnetic (EM) wave interacts with the surface in a way that depends from the working wavelength (λ): only objects with size comparable toλreact to the EM illumination. Every scatterer should be known within a small fraction of a wavelength (usually centimeters) in order to describe completely the imaged scene. This requirement never met for natural distributed scenes like rough surfaces. Hence the SAR images are preferably treated as random processes.
Generally the resolution cell has a greater order of magnitude with respect to the wavelength (for example for the ENVISAT system the azimuth-range resolution is 4×20mand a wavelength of 5.6cm), hence it is realistic to assume that within the resolution cell many elementary scattering mechanisms happen. This is the main assumption that leads to the formulation of a statistical model for the SAR SLC. The monochromatic nature of the SAR implies that all the elementary scattering mechanisms interfere between them in a constructive or
destructive way. The value of the pixel is the coherent sum of these elementary contributions and it is also indicated as random walk to
refer at the representation of the sum in the complex plane as shown in Fig. 1.7.
Figure 1.7: Random walk
A pixel value results from the superimposition of several interference phenomena. As a consequence, the pixel can differ in value from the neighboring pixels even for homogeneous areas (phisical property constant in space). This result is evident on the SLC’s amplitude as a
salt and pepper disturb and it is known asspeckle effect [14, 81]. This
phenomena is present in any sensing technique that uses a coherent receiver as, for example, ultrasound, laser optics, etc..
Since the scattering mechanism can be seen as a summation of N discrete contributions, the integrals in 1.1 can be replaced by summa-tions. If we consider also its representation in the complex plane:
u(x, r) =PN
n=1u(x0, r0)eiφ(x
0,r0)
sinc(x0−x)sinc(r0−r)dx0dr0 = =M(x, r)·ejφ(x,r)=< {u}+j= {u}
(1.20) where<and =indicate respectively the real and the imaginary part, one can rewrite the expression separately as:
u(x, r) = N X n=1 Dnejφn (1.21) < {u}= N X n=1 Dncosφn (1.22) = {u}= N X n=1 Dnsinφn (1.23)
The statistical description of a SAR SLC is based on the following hypothesis [14, 51]:
• the amplitude and the phase of the elementar scatterers are statistically independent of each other and from the amplitude and phases of all other elementary scatterers;
• the phases of the elementary scatterer can lie with same proba-bility anywhere in the interval (−π, π(.
The first assumption is accomplished given that the propagation phase delay is independent from the scattered wave strength. The second, instead, comes directly from the fact that coherent summation of in-correlated scattering mechanism results in random phase values that are uniformely distributed in (−π, π( once folded in that interval. Nor-mally the hypothesis of a large number of elementary scatters in the
resolution cell is made. Under the assumption that the fuctions of < and = respect the central limit theorem, we know that they are distributed as a Gaussian probability function with zero mean:
E{< {u}}= N X n=1 E{Dncosφn}= N X n=1 E{Dn}E{cosφn}= 0 (1.24) E{= {u}}= N X n=1 E{Dnsinφn}= N X n=1 E{Dn}E{sinφn}= 0 (1.25)
and with variance:
En<2{u}o= N X n=1 EnD2noEncos2φn o = N2 ·EnD2no (1.26) En=2{u}o= N X n=1 EnD2noEnsin2φn o = N2 ·EnDn2o (1.27)
where E stands for the statistical mean operator. Furthermore one has to notice that real and imaginary part are incorrelated:
E{< {u} = {u}}= N X n=1 N X m=1 E{DnDm}E{cosφnsinφm}= 0 (1.28)
scatterers. Renaming withx and y the real and imaginary part: px(x) = √ 1 2πσ2e −1 2( x σ) 2 x(−∞,∞) (1.29) py(y) = √ 1 2πσ2e −1 2( y σ) 2 y(−∞,∞) (1.30)
where the σ is the standard deviation computed from formulas 1.26 and 1.27. Both distributions are gaussian with zero-mean and same variance hence the backscattered field u is defined as a circular com-plex gaussian process.
From the last relationship on real and imaginary part it is possible to derive the pdf of amplitude M =px2+y2 while the pdf for the phase is a uniform distribution. The phase and amplitude pdfs are separable. pM(M) = M σ2e −M2 2σ2 M [0,∞) (1.31) pφ(φ) = 21 π φ[−π, π) (1.32) The amplitude is distributed as a Rayleigh while the phase is uniform and hence it doesn’t depend on the scatterer, that means that the phase of the SLC doesn’t bring any information. The Rayleigh pdf is described by only one parameter: its standard deviation σ. In fact the mean and the variance are respectively given by: E{M}=σpπ/2
and σM2 = (2−(π/2))σ2
The intensity is instead distributed as an exponential random variable, defining I =M2 :
pI(I) = 1
2σ2e
− I
2σ2 I[0,∞) (1.33)
with equal mean and variance: E{I}=σI2 = 2σ2.
A target that meet the property of N independent scatterers, with no scatterer that remarkably dominates the others is definedgaussian scatterer (for the reason we have already seen) ordistributed scatterer
and the model described before holds [14, 109]. For medium resolution (tens of meter) SAR images, this description is met for most of the natural scatterers such as forests, agricultural fields, rough water, soil or rock surfaces. In the literature, the speckle resulting from the imaging of a distributed scatterer is defined as fully developed speckle
[14, 70, 109]. This condition is violated when one or few scatterers are predominant with respect to the others in the resolution cell as it happen for artificial objects, urban areas or with very-high resolution SAR acquisitions. These last situations refer to the point scatterer
case in which the pixel value is deterministic and the speckle is defined aspartially developed speckle [14, 70, 109].
2.1 Overview
In the previous chapter the Synthetic Aperture Radar and its pecu-liarities have been presented. A single SAR image has been described as a plane representation in the azimuth and range directions of the three-dimensional observed scene. When also the height of the imaged scene is of interest, a different application is needed. As for the Stere-oscopy, in which two different views are necessary in order to achieve the optical depth information, also in the SAR vision two acquisitions are needed in order to acquire the heigth information. If two SAR im-ages acquired from two slightly different view angles are considered, their phase difference can be usefully exploited to generate Digital El-evation Maps (DEMs) and/or monitor terrain changes (deformation velocity map) [14, 52].
Under the name ofSAR Interferometry (InSAR) are indicated all the
methods that employ at least two complex-valued SAR images, indi-cated as interferometric pair, to derive additional information about
the sensed target by exploiting the phase of the SAR signals. De-pending on the information that has to be estimated, the two SAR images have to differ at least for one imaging parameter. Which pa-rameter this is, determines the type of the interferometer, e.g. flight path for across-track interferometry, acquisition time for along-track interferometry (ordifferential interferometry - DInSAR), etc. [3].
Interferometric SARs can be operated from aircraft or satellite. The latter provide data that are at least one order of magnitude cheaper than airborne data and this is particularly true for inaccessible areas of the Earth. Aircraft SAR however ensures high flexibility that is of primary importance in emergency situations, when SAR products are required as quick as possible.
Figure 2.1: SAR interferometry processing chain
Fig. 2.1 shows the InSAR processing chain starting from the two fo-cused SAR images. Since the real scene has been imaged from two different view angles, the position of the scene in the two images is normally different. So, before any other processing the two images
have to be aligned. In the specific, the image taken as reference is defined asmaster and theslaveis morfologically transformed in order
to be aligned to the master. This operation, that goes under the name of coregistration,is a crucial step in the SAR processing chain and it
can hardly influence the system performances. Multiplying the master SLC by the complex conjugate of the slave SLC, the complex inter-ferogram is obtained. In this step, a filtering procedure is normally
exploited in order to mitigate the interferometric phase noise. This operation is normally indicated ascomplex multilooking and improve
the phase reliability at the cost of spatial resolution impairment. In the next chapters, starting from a statistical description of the interfer-ometric phase, the complex multilooking will be extensively discussed in addition to other interferometric phase estimation methods. Since the received phase is measured in the interval (−π, π(, there ex-ist an ambiguity of the phase to within integer multiples of 2π due to the wrapping of the absolute phase values in the above mentioned in-terval. The 2πcycle of phase is indicated asinterferometric fringeand
the procedure that allows to obtain the absolute values of ψfrom the measured phase is stated asphase unwrapping. In order to obtain the
absolute height value some ground reference points of known altitude (generally given by mean of corner reflector) are needed.
Eventu-ally a procedure ofgeocoding is necessary to correctly positioning the
measured height values on the earth surface.
2.2 Across-track Interferometry
The Digital Elevation Model generation has been one of the first goal that brought to the development of the SAR interferometry technique. This is accomplished in the so called across-track interferometry that
is an InSAR configuration that resembles to a stereo arrangement: two SAR sensors fly on ideally parallel tracks and view the terrain from slightly different directions [52, 87, 90, 119]. Across-track InSAR is a mean to measure the elevation angle θ as a third coordinate, beside azimuth and range, and it allows thus to recover the point’s location in space. Fig. 2.2 shows the across-track geometry in the (y,z)
cross-section.
Depending from the way the two images are acquired, the across-track configuration is indicated as single-pass interferometry, when the
im-ages are acquired at the same time by means of two radars flying simultaneously, or alternatively repeat-pass interferometryif only one
radar is used and the scene is imaged in two different satellite pas-sages. In this last case, the time between the two acquisitions, named astemporal baseline, strongly affects the quality of the InSAR
acquisi-tion. As it will be shown extensively in the following, also the spatial distance between the radars at the time of acquisition, the spatial baseline, is even a more important parameter that defines the quality
of the interferometer [12, 60].
As one can see in Fig. 2.2, for a generic pointP on the ground surface,
the information on the heightz is codified in the difference4r of the electromagnetic paths related to the 2 acquisitions.
Letu1(·),u2(·) be the two focused SAR images:
u1(x, r) =´ ´ γ(x0, r0)e−j4λπr 0 sinc[a(x0−x)]· ·sinc[b(r0−r)]dx0dr0 (2.1) u2(x, r) =´ ´ γ(x0, r0)e−j4λπ(r 0+4r) sinc[a(x0−x)]· ·sinc[b(r0−r− 4r)]dx0dr0 (2.2)
where the pair (x, r) refers to the image coordinate and (x0, r0) refers to the observed scene coordinate.
Assuming the image u1(·) as the master SLC, under the hypothesis of perfect coregistration and independency of 4r from x and r, the coregistered image u2(·) is:
u2(x, r+4r) =ρc(x, r)·u1(x, r)·e−
4π
λ4r (2.3)
where the quantity ρc is a complex quantity, defined as complex
co-herence, that takes into account the possible decorrelation phenomena between the master and slave SLCs and will be in the following exsten-sively described.
SLC by the complex conjugate of the slave SLC:
Γ(·) =u1(·)u∗2(·) =|u1(·)||u2(·)|ejψ(·) (2.4)
withψ(·) =φ1(·)−φ2(·) defined asinterferometric phase:
ψ=m·k· 4r=m2π
λ4r (2.5)
wherek is the wavenumber andm is an integer number that depends
from the across-track configuration. In particular m = 1 when the path difference depends only from the return path (single-pass con-figuration), instead m = 2 when the path difference accounts for the trasmission plus receiving path (repeat-pass configuration).
Figure 2.3: InSAR acquisition geometry in the (y,z) plane.
Assuming a repeat-pass configuration and referring to Fig. 2.3, the quantity 4r is derived from the system geometry. Considering the
triangle between the points P1−P2−P, the following relation holds:
(r+4r)2 =r−Bk
2
+B2⊥ (2.6)
whereBk and B⊥ are respectively the parallel and the perpendicular baselines, whose expressions are linked to the spatial baseline B as:
Bk =B·sin(θ−α) (2.7)
B⊥ =B·cos(θ−α) (2.8)
Observing that (B,4r) r, an approximation for the term 4r is derived from eq. 2.6 by neglecting4r2 with respect tor2 andB with respect tor:
4r∼=B·sin(θ−α) (2.9)
and consequently, from eq. 2.5, a value forψ:
ψ= 4π λ · 4r
∼
2.2.1 Double differences
The derived phase in eq. 2.10 is the measured phase in one pixel. By itself, this quantity is not useful bacause it still contains the range information and because the phase is wrapped. This means that the measured interferometric phase cannot be taken in an absolute sense, but it should be meant as a differential measurement. In fact, in order to remove the range information in the interferometric phase measure-ment, the relative phase with respect to a reference point in the image is normally considered. This is the reason of why often this mecha-nism is described as double differences: the first is the one between
SLCs in the interferogram generation step at the aim of removing the intrinsic phase of the target: the φ0 contribution in eq. 1.2. The second difference is done between interferometric phases in order to remove the range absolute path dependence.
If the interferometric phase difference between two adjacent pixels P and P’ is considered:
4ψ= 4π
λ · 4(4r) (2.11)
From the geometry in Fig. 2.4, the variation4(4r) from the point P to P’ can be determined. In particular, considering the variation in the look angle 4θ, the previous equation can be written as:
4ψ= 4π
λ [B·sin(θ−α)−B·sin(θ+4θ−α)] (2.12) Expanding the second term on the basis of the addition formula for the sine function assin(θ+4θ−α) =sin(θ−α)·cos(4θ)+cos(θ−α)·
Figure 2.4: InSAR double differences principle.
sin(4θ) and considering the fact that 4θ 1 since the two pixels are adjacent, the approximation sin(4θ) ∼=4θ can be done and the last equation become eventually:
4ψ= 4π
λ4θB·cos(θ−α) = 4π
λ4θB⊥ (2.13)
In order to derive the expression of 4θfrom the geometry in Fig. 2.4, it should be noted that it depends from two contributions, one due to the range difference between acquisitions (4r) and another one due to the height difference (4z). This two contribution can be separated and indicated respectively asflat-earth and topography contributions:
4ψ=4ψf lat+4ψtopo (2.14)
For the first contribution, under the usual assumption of 4θ 1, it comes that r·sin(4θ)∼=r· 4θ=4r/tan(θ) and the equation 2.13
can be written as: 4ψf lat= 4 π λ B⊥4r r·tan(θ) (2.15)
That means that the flat earth generates a linear interferometric phase pattern.
The topography component is derived under the assumption of no range difference but only height difference and from the usual geome-try it resultsr·sin(4θ)∼=r· 4θ=4z/sin(θ), obtaining eventually:
4ψtopo = 4π λ
B⊥4z
r·sin(θ) (2.16)
This last equation shows the relationship that holds between the height variation and the interferometric phase variation between adja-cent pixels. When the aim is to measure the topography (DEM), the flat-earth contribution is removed and a phase-to-height conversion is done. The derived height information is relative to a reference point, as explained above, hence ground-reference points of known altitude are necessary in order to obtain the absolute height measurement of the imaged scene.
2.2.2 System sensitivity and critical baseline
Equation 2.16 states the capability of the system to measure height variations, in other words, the interferometer sensitivity to the
4z: 4ψ 4z = 4π λ B⊥ r·sin(θ) (2.17)
It is evident that the system sensitivity depends mainly from the or-togonal baseline B⊥ and from the λ. While the λ is generally fixed depending on the application, the ortogonal baseline is a parameter that can be set in accordance to design constraints as for example in the DEM generation application. In order to improve the system sensitivity, the orthogonal baseline can be increased: the farther the two satellite paths are, the more sensible the system is. At the same time an higher sensitivity means that, for a same slope, the phase wrapping is more frequent (higher fringe density), leading to a more difficult phase unwrapping and height information extraction. This behavior leads to interferometric phase impairment and it is indicated with the name of baseline decorrelation. The limit to the maximum
orthogonal baseline that causes complete decorrelation is indicated as
critical baselineand it is defined as the baseline that causes a 2πjump in the interferometric phase for a unitary slant-range variation of the topography. The equation for the critical baseline can be derived from equation 2.15 by setting 4ψ 4r = 2π: 4ψ 4r = 4π λ B⊥ r·tan(θ) = 2π (2.18) and eventually: B⊥,critic= λ 2r·tan(θ) (2.19)
Normally, in the generation of the DEM, the useful baseline value is far away from the critical baseline. By substituting the relative operating parameter, the ENVISAT system has a critical baseline of circa 9km.
2.2.3 Height of ambiguity
If the same reasoning made for deriving the critical baseline is also made starting from equation 2.16, it is possible to derive an important parameter for the interferometric system. So imposing 4ψ = 2π in equation 2.16 and deriving the height it results:
zamb=
λr·sin(θ)
2B⊥ (2.20)
This parameter expresses the height variation that generates a phase change of 2πor, in other words, the height that is encoded in one inter-ferometric fringe and is defined asheight of ambiguity. This parameter
expresses something similar to the system sensibility in eq. 2.17. The more sensitive is the system, the smaller the height of ambiguity is. Generally in DEM generation campaigns the height of ambiguity is a requirement on the final product. In order to achieve the desired min-imum requirements often multiple acquisitions are made and only the one that respect all the requirements on SNR and height of ambiguity is chosen for the phase to height conversion and DEM generation.
2.2.4 Spectral shift
Given that the operating orthogonal baseline should be much smaller than the critical baseline value, under this limit there is another
degra-dation due to the diversity of the view angles that can also be ac-counted as baseline decorrelation. In [44, 88] it has been shown that the spectra of the SAR images acquired from two different view an-gles are different bands of the ground reflectivity’s spectrum. In fact, with reference to Fig. 2.4, supposing that the slope from P to P’ is
constant and form an angleβ with respect to the ground, the ground-range wavenumber is:
ky = 4
π
λsin(θ−β) = 4πf
c sin(θ−β) (2.21)
By differentiating with respect toθ, the variation in the ground-range wavenumber 4ky related to a variation 4θ in the look-angle is
ob-tained:
4ky = 4
πf4θ
c cos(θ−β) (2.22)
A variation in the look-angle causes a shift and a stretch of the im-aged terrain spectra. The shift is due to the term4θand the presence of f gives a not uniform translation along the frequency (frequency-dependent shift) and than stretches the spectra. This behavior is defined in [44] as thewavenumber shift orspectral shift. By imposing
that the system bandwidth is small with respect to the central fre-quency f0, it is possible to ignore the stretch by approximating the previous formula as:
4ky = 4
πf04θ
This equation indicates that two SAR surveys acquire two different bandwidth of the ground reflectivity’s spectra in dependance from the acquisitions’ look-angle. The frequency shift between the two acquired spectra is derived by differentiation of eq. 2.21 with respect tof:
4f =− cB⊥
rλtan(θ−β) (2.24)
Because of the wavenumber shift the two SAR signals are not fully correlated. As seen in the last equation, this decorrelation is higher as larger the orthogonal baseline is and, for this reason, it is accounted as baseline decorrelation. In fact it is worth noting that by imposing4f equal to the system bandwidth, the critical baseline is again obtained. In order to reduce the decorrelation between the two SAR signals a band-pass filtering, that limits the two spectra to the common band, is performed before the interferogram generation. This operation is named common band filtering and described in details in [44].
2.3 Differential and along-track Interferometry
As shown in the previous section, one way to perform across-track interferometry is by means of repeated passes from the same radar and image the scene in two different times. If a ground deformation occurs, and this is the case of subsidence phenomena, earthquakes, landslides, etc., there is a further contribution that appears in the interferometric phase expression. In order to detect the correct to-pography, this quantity should be measured and taken into account. When instead the deformation phenomena is the quantity to mea-sure, on-pourpose systems are designed as in the cases of Differential
interferometry (DInSAR) and along-track interferometry. This de-formation contribution is independent from the spatial baseline and
depends only from the operating wavelength and the projection on the slant-range direction of the occurring deformation (d):
ψdef = 4
π
λ d (2.25)
In along-track interferometry the two SLCs are acquired by means of two antennas aligned with the fligth path or two satellites flying within the same orbit. The absence of a spatial diversity between the points of view make the topographic contribution being null and only the deformation contribution arises. In the general case of no perfect alignment along the orbit it comes again an across-track configuration, the application goes under the name of Differential InSAR and both topographic and deformation contributions are present. This is the most general case since it is difficult to obtain perfectly aligned orbits between acquisitions.
Considering Fig. 2.5 and the movement dalong the slant-range direc-tion (from the point P to Pd). Said S1 and S2 the two satellites
po-sitions, B⊥ the orthogonal spatial baseline,rs andrsd the slant-range
distances of the target positionsP andPdfromS2, the interferometric
phase is:
ψ= 4π λ rsd−
4π
λ r (2.26)
The previous expression can be rewritten so that the deformation d and topography 4r terms are separated:
λψ 4π =rsd−rs+rs−r=d+4r (2.27) and hence: ψ=ψtopo+ψdef = 4 π λ 4r+ 4π λ d (2.28)
By removing the flat-earth component, from eq. 2.16 it results:
4ψ= 4π λ B⊥4z r·sin(θ) + 4π λ d (2.29)
As previously said, it should be noted that, in constrast with the to-pographic contribution, the deformation term does not depend from the spatial baselineB⊥. The last equation is important since it shows that a variation of d of half the wavelength can bring a 2π jump in
the measured interferometric phase. This means that the system is much more sensitive to deformations than to the topography. In order to have an idea of the order of magnitude of the detectable deforma-tion, the ENVISAT system parameters (taking a value of orthogonal baseline of 150 m) are substituted and it results:
4ψ= 4z
10 + 225d (2.30)
The higher sensitivity to the deformation can be noticed in the dif-ference of the terms that multiply the quantities 4z and d. For the ENVISAT system then the sensitivity is 2.8 cm.
The deformation term can be separated from the topographic one in different ways:
1. the pair is acquired with a very small baseline so that the topog-raphy contribution is zero as in the along-track interferometry. 2. If an accurate DEM is available, the topography can be
esti-mated and subtracted from the measured phase. The remaining phase will account for the deformation and the unavoidable er-rors on the DEM.
3. With three SLCs, a DEM can be generated from a pair and used as in point (2).
4. A set of coregistered SLCs is available and a model for topog-raphy and deformation velocity can be set at the aim of the estimation of a constant rate of displacement or a singular mo-tion event.
Other contributions to the interferometric phase should be taken into account. In a realistic scenario, the interferometric phase will also
account for the following terms:
ψ=ψtopo+ψdef +ψprop+ψscat (2.31)
where the single terms are:
• ψtopo is the topography contribution;
• ψdef accounts for a possible displacement of the scatterer
be-tween observations;
• ψpropis a possible phase delay difference due to ionospheric and
atmospheric propagation conditions: tropospheric water vapour and rain cells are dominant sources for this phase error;
• ψscat stands for the influence of any change in the scattering
behaviour.
Propagation medium effects ψprop enter the DInSAR measurement
directly, it can only be suppressed when a multi-temporal dataset is available and averaging several observations. Instead a phase shift ψscat due to changes in the scattering properties cannot be
distin-guished from the deformation term and will affect the estimation.
2.3.1 Multi-pass Differential Interferometry
The advent of new satellite systems with a better control of fligth or-bits has made possible the acquisition of set of images related to the same area. This allows to observe a same area over a long time span and model the deformation along the time. Multi-pass DInSAR tech-niques start from a dataset of N images of the same scene, acquired at different time t1, t2, . . . , tN. A number of M interferograms are
generated, where M ≤N(N −1)/2. Each interferogram has its own temporal and spatial baseline, that are organized in the multitem-poral/multibaseline stack (MT/MB). One of the possible strategies aimed for making the choice of M interferograms can try to minimize the temporal and spatial baseline in order to reduce temporal and spatial decorrelation respectively.
Let us approximate the trajectory as straight in the observation time so that the deformation can be expressed asd(ti, P) =v(P) (ti−t0),
wherev(·) is thevelocity of deformation. Not considering the disturbs
on the interferometric phase, only the topographic and deformation contributions remain and the phase of the i-th interferogram related to a pixel P can be expressed as:
ψi(P) =−4 π λ B⊥i r sinθz(P) + 4π λ v(P) (ti−t0) (2.32) For each point P then we have a linear term in thebaseline dimension
and one linear term in the time dimension. In order to estimate the
topography and the velocity of deformation, a model for the interfer-ometric phase is assumed. Generally this is a bidimensional complex exponential whose frequencies are proportional to topography and de-formation:
s(i, Bn;P) =ejψ(i,Bn;P) (2.33)
The ML estimates of topography and velocity (z, v) is then the max-imization of the 2D periodogram in thebaseline-time domain [29, 32,
As seen in the previous paragraph, the propagation contributionψprop
cannot be easily removed from a single interferometric acquisition. This contribution, mainly due to atmospheric effects, causes different propagation delays along the image. When a multi-pass dataset is available, the atmospheric disturb can be neglected by incoherently averaging it over the different acquisitions.
PS technique
The first approach that has shown the effectiveness of the use of multiple interferometric surveys for terrain deformation monitoring is the Permanent Scatterer Interferometry (PSI) technique proposed
in [29, 30]. This approach aims to estimate the pair (z, v) only on specific reliable targets, named permanent scatterer (PS),
character-ized from extremely low temporal and spatial decorrelation. Such a kind of scatterer can be found in mountainous and urban regions as rocks and man-made structures. Choosing targets that do not suffer from baseline decorrelation implies that the processing does not need any pre-estimation of interferometric phase and coherence, allowing in this way a full resolution processing.
In this approach one SLC is chosen as master image and all other interferograms are generated with respect to it. A set ofM non-linear
equation results by generating M =N −1 interferograms.
While no processing to estimate the phase is not needed, the pre-liminar and crucial step of PSselectionshould be carefully performed.
Since the phase is affected by several disturbs, in order to find the PS only the pixel amplitude is considered over the N available SLCs and
the most stable pixels are choosen.
This technique shown for the first time the capability of neglecting the atmospheric disturb, i.e. the 4φprop term in eq. 2.31, by means
